Abstract

Intravenous injection of nanoparticles as drug delivery vehicles is a common practice in clinical trials of therapeutic agents to target specific cancerous or pathogenic sites. The vascular flow dynamics of nanocarriers (NCs) in human microcapillaries play an important role in the ultimate efficacy of this drug delivery method. This article reports an experimental study of the effect of nanoparticle size on their margination and adhesion propensity in microfluidic channels of a half-elliptical cross section. Spherical polystyrene particles ranging in diameter from 60 to 970 nm were flown in the microchannels and individual particles adhered to either the top or bottom wall of the channel were imaged using fluorescence microscopy. When the number concentration of particles in the flow was kept constant, the percentage of nanoparticles adhered to the top wall increased with decreasing diameter (d), with the number of particles adhered to the top wall following a d−3 trend. When the volume concentration of particles in solution was kept constant, no discernible trend was found. This experimental finding is explained by the competition between the Brownian force promoting margination and repulsive particle–particle electrostatic forces retarding adhesion to the wall. The 970 nm particles were found to adhere to the bottom wall much more than to the top wall for each of the three physiologically relevant shear rates tested, revealing the effect of gravitational force on the large particles. These findings on the flow behavior of spherical nanoparticles in artificial microcapillaries provide further insight for the rational design of NCs for targeted cancer therapeutics.

FIGURES IN THIS ARTICLE

Introduction

Conventional cancer drug delivery approaches often rely on exposing a lethal number of healthy cells to the cytotoxic drugs used in chemotherapy resulting in myelosuppression, immunosuppression, and all too often death. Recent interest in a modality for targeted drug delivery preferentially to the site of diseased cells based on rational design of drug-loaded nanoparticles may offer promise. These drug-loaded nanoparticles or nanocarriers (NCs) should move periodically toward vessel walls in flow, recognize and adhere to diseased cells, and deliver drugs preferentially inside of targeted cells [1]. Their potential for engineered site-specific delivery make NCs a promising medicament to the side effects associated with traditional cancer therapies [2].

For spherical NCs, a myriad of factors can affect the ultimate drug delivery efficacy, owing the many variables to the complexity of delivery environment. These factors include particle size, surface properties, as well as dynamic vascular conditions [3-5]. In order for a NC to migrate from the injection site through successively smaller blood vessels into microcirculation, tumor vasculature, and eventually adhere to a diseased cell, it must tend to marginate radially toward the endothelial walls in vascular transport [6]. This propensity to marginate and adhere to vascular walls is an important factor in rational NC design and has been shown to be desirable in improved NC delivery efficacy [1].

In their journey through the circulatory system, there exist a plethora of barriers to successful delivery of NCs to diseased sites associated with their size. The first of these barriers is derived from the fact that diseased endothelia are characterized by gaps or fenestrations between adjacent cells as large as 500 nm. This physiological anomaly is exploited in a particle delivery strategy known as the enhanced permeation and retention effect [7]. The strategy is to deploy NCs on the order of 500 nm and smaller. These nanoparticles can extravasate preferentially through the larger voids of diseased tissue, but cannot pass through the relatively contiguous endothelial cells present in healthy vessel walls. Additionally, due to underdeveloped lymphatic drainage, particles can remain in the tumor environment for an extended period of time. The second size barrier to successful NC delivery comes from the Mononuclear Phagocyte System (MPS), which is the body's natural filtration system for removing potentially harmful particulate from circulation [8]. The MPS is comprised of cells, mainly macrophages and monocytes lining the sinusoids of primarily the liver and spleen. Particles of characteristic size greater than 200 nm have been shown to accumulate preferentially in the organs of the MPS in vivo [9] and coating NCs with polyethylene glycol results in longer circulation times [10]. Finally, upon successful delivery to a target cell, the optimum spherical NC size for cell internalization through receptor-mediated endocytosis has been shown to be on the order of 50 nm both theoretically and experimentally [11,12]. However, the NC must have a diameter larger than approximately 10 nm to avoid rapid renal clearance [13].

The effect of spherical polystyrene particle diameter on margination and adhesion propensity has been investigated fairly extensively in conjunction with parallel plate flow chambers (PPFCs). PPFCs encompass a broad class of experimental setups. In general they consist of a glass surface, often with a relevant cell culture, bonded to a polydimethylsiloxane gasket to define a rectangular cross section flow cell such that the flow field can be considered two-dimensional. The number of spherical particles adhered to the bottom wall of PPFCs has been characterized for diameters ranging from 20 μm down to 50 nm with varying experimental conditions. Decuzzi et al. studied the adhesion and margination behavior of spherical particles with diameters between 10 μm and 500 nm in a PPFC of height 254 μm [14]. At a shear rate of 7.75 s−1 and constant volume concentration, the percentage of particles marginating and adhering increased with increasing particle diameter (d) as d1.3 [14]. In a different study, margination of polystyrene particles larger than 500 nm in diameter to the bottom wall was attributed chiefly to the gravitational force [15]. Based on a theoretical calculation, the margination of large particles is driven mainly by gravity with the volume adhered (Va) depending on Cd4, where C is the local number concentration of particles. This result is equivalent to a linear diameter dependence in the percentage of particles adhered to the wall. In one experiment, keeping the total volume of particles injected constant by setting the injected particle concentration (C0) proportional to d−3, the authors observed that the volume adhered [16] or the percentage of particles adhered was proportional to d1.25, approximately in agreement with the theory. For another experiment using particles of diameters between 50 and 200 nm at constant volume concentration, the volume deposited was found to be proportional to d3.2 for constant C0. Because Va is predicted to scale with d4 when C is constant and gravity is the dominant margination force, it was suggested that other forces play an increasingly important role in the margination of the sub-200 nm particles [15]. Due to the nature of the PPFC geometry, all of these relations are for particles sedimenting and adhering to the bottom surface of large flow chambers relative to human capillary diameters. In comparison, the percentage of particles injected at constant number concentration adhered to all four walls of a rectangular cross section PPFC with dimensions of 175 × 100 μm2 measured using overall fluorescence was shown to increase with decreasing particle diameter between 135 and 60 nm in a study reported by Toy et al. [17]. This result suggests that margination behavior to the bottom wall may be different from those to the other walls; however, Toy et al. did not quantify the different margination behaviors to individual walls.

The aforementioned studies were conducted at relatively low shear rates. Patil et al. examined the additional effect of spherical particle size on adhesion of large particles (5–20 μm) and found the existence of a critical shear rate, above which adhered particles are sheared off of the wall. The critical shear rate was found to increase with decreasing diameter for constant number concentration [14,18,19]. However, at a sufficiently low shear rate of 75 s−1, the percentage of particles adhered was shown to be independent of diameter, different from the trends discussed in the preceding paragraph.

The decrease of apparent viscosity of blood with decreasing channel diameter is known as the Fahraeus–Lindqvist effect [20]. This drop in apparent viscosity is the result of lateral migration of red blood cells (RBCs) resulting in a two-phase flow, i.e. a RBC core and an RBC-free plasma layer adjacent to the wall known as the cell-free layer. In vessels where RBCs are present, the RBC-dense core of the flow acts to push microparticles out of that region toward the vessel walls, trap NPs in the core, and disrupt NP diffusion across the core [18,21,22]. The extent to which RBCs trap NPs, dispel microparticles, and prevent diffusion depends on particle size and the number of RBCs present, which is a function of channel diameter. Abbitt and Nash reported that larger neutrophils adhered more efficiently in a 300 μm PPFC than smaller lymphocytes when flown in whole blood [23]. These in vitro results were observed in PPFCs. However, the RBC migration and the fluid flow behaviors are profoundly influenced by bifurcations and other complex geometries that characterize in vivo vasculature [24]. Nevertheless, it has been shown both theoretically and experimentally that as the vessel diameter decreases, the cell-free layer thickness decreases until RBCs are eventually excluded from flowing in microcirculatory vessels [25,26].

In experiments performed in model blood flow, Charoenphol et al. reported that for particles ranging from 100 nm to 10 μm adhesion generally increased for increasing particle diameter, wall shear rate, and channel height [18]. The shear rate has been shown to be important in the overall margination characteristics of large particles in other studies as well [27]. The channel height has also been shown to affect large particle margination and adhesion propensity in multiple studies. Specifically the percentage of large particles adhered to the bottom wall of a PPFC was shown to increase with increasing channel height for PPFC gasket height ranging from 127 to 762 μm, which is still larger than the 20–50 μm range for microcirculatory vessels [18].

Although the aforementioned experimental studies have revealed size effects of particle margination and adhesion behaviors, there is still a lack of clear understanding of the origin of the different trends observed. Here we report an experimental study that quantifies the different margination behaviors of 60–970 nm diameter polystyrene spheres to the top and bottom walls of microfluidic channels with a radial dimension of 35 μm, comparable to human venules and capillaries, under physiologically relevant shear rates. Few RBCs are expected to be present in microcapillaries of a similar radial dimension. Hence, our experiments have been conducted in a RBC-free solution. We have found that the 970 nm particles marginate to the bottom wall much more than to the top wall due to gravity-driven sedimentation. In comparison, smaller particles exhibit increasingly uniform margination due to Brownian diffusion. For a constant particle concentration, the percentage of these submicron polystyrene spheres that marginate and adhere to the top channel wall increases with decreasing particle diameter down to at least 60 nm and follows a d−3 relation, suggesting that the same volume is delivered regardless of the diameter of the sub-900 nm particles. However, for constant volume concentration of particles in the flow, a clear diameter dependence of percentage of particles adhered was not observed. These results as well as the different trends reported in the literature are explained by examining the competition between the gravity, Brownian, hydrodynamic, and interparticle electrostatic forces. Based on our analysis, the repulsive electrostatic force between adhered particles and particles in the flow, a factor that has not been emphasized in past studies, can play a large role in the size dependence of particle margination and adhesion behaviors.

Methods

Half-elliptical cross section microfluidic channels were fabricated in soda lime glass wafers using photolithography and etching techniques. An isotropic wet etch defining the channel geometry produced a half-elliptical cross section with a 55 μm major axis and 35 μm minor axis, as shown in Fig. 1(a). Fluorescent polystyrene nanospheres of diameter 970, 780, 390, 210, 93, and 60 nm presenting surface carboxyl groups (Bangs Laboratories, Fishers, IN) were used as model NCs. The zeta potentials of the 60, 93, 210, 390, 780, and 970 nm polystyrene particles in DI water were measured to be −40.4, −46.4, −45.5, −41.2, −38.1, −44.0 mV, respectively, with ±4.2 mV uncertainty for each value. Particles were flown at either a constant particle concentration of (1.0 ± 0.2) × 109 particles/ml or at constant particle volume fraction of 3.1 ± 0.6 vol. %. The particle solution was flown at a flow rate of 15 μl/min for all studies except for the flow rate dependent study, where particles were flown at 15, 25, or 35 μl/min through the straight glass channel. Because the viscosity of blood flow is higher than that of the DI water used in our study, these flow rates are larger than RBC flow rates reported in the literature, which vary from 0.03 μl/min [28] to 2 μl/min [29] for vessels of similar sizes, to obtain a comparable wall shear stress. With the viscosity of whole blood taken to be 3.5 mPa·s [30], the shear stress in real blood vessels can be calculated from the Hagen-Poiseuille equation to be on the order of 8.4 Pa [29]. The shear stresses for the current study are estimated using the following equation from Bahrami et al. [31]

where τ is the mean wall shear stress, μ is the dynamic viscosity, c is the minor axis of the half-ellipse, ε is the ratio of the minor to the major semi-axes, U¯ is the mean velocity, A is the cross-sectional area, and P is the perimeter of the channel. For the three flow rates used in the study, the calculated shear stress values are 13, 22, and 30 Pa.

Images were taken of the top or bottom wall before the flow, at 1 s, and every 20 s for a total flow time of 360 s. Representative florescence images of 93 nm particles adhering to the channel's top wall at 20 and 360 s are shown in Fig. 1(b). For each image obtained for particles larger than 100 nm the number of particles adhered to the top or bottom wall of the 55 × 120 μm2 imaged region of the microchannel were counted. For 93 and 60 nm particles, where the number of particles adhering became very large, a particle quantification technique based on individual particle counting and fluorescence intensity was employed. A region of interest 55 × 17 μm2 on the channel wall was designated as indicated by the yellow rectangle in Fig. 1(b). For the first three images, taken at 0, 20, and 40 s, fluorescence intensity profiles were gathered. We then calculated the average intensity of the region of interest based on the measured intensity distribution. For 93 and 60 nm particles were counted for the 20 and 40 s time points. The average intensity at 20 s was subtracted from the average intensity at 40 s to obtain an intensity gain. We then calculated the ratio between the intensity gain and the number of additional particles adhered between the two time points. This ratio was used to determine the number of additional particles adhering for the remaining time points based on the measured average intensity gain. Figures 1(c) and 1(d) show the time dependent intensity profiles along with the correlated new particle adherence for the run of 93 nm particles. The number of particles in the 55 × 17 μm2 region of interest was then multiplied by seven to account for the 55 × 120 μm2 full image area in order to maintain consistency with larger particle data. For data where the volume concentration of particles in the flow was maintained constant, a volume concentration of 3.1 ± 0.6 vol. % was used in order to retain the particle concentration for 390 nm particles as a benchmark.

In order to verify the accuracy of the fluorescence intensity counting method, it was used to count the number of particles in a run of 390 nm particles for 20 and 40 s and then compared with the manual counting method for all time points. This was done because 390 nm diameter particles are readily distinguishable as individual particles for all time points. Figure 2 shows the agreement between the two methods for one run of 390 nm particles.

In switching between imaging particle adherence to the top or bottom wall, the channel was cleaned using acetone, air, then water and inverted so that the flat surface was placed in the desired orientation. When the flat surface was the top surface, an upright Olympus BX-41 fluorescence microscope was used for imaging particles adhering to the flat surface. When the flat surface was the bottom surface, an inverted Ziess Axiovert 200 M fluorescence microscope was used to image particles on the flat surface. The zeta potential of each particle size was measured in DI water using a Zetasizer Nano ZS (Malvern Instruments Ltd.).

Results

Figure 3(a) shows a typical image of the top wall at 360 s for each of the particle sizes tested. In each case the trial began at t = 0 s with no particles adhered to the walls. At t = 1 s, a number of particles adhered to the wall due to transient capillary pressure and wetting phenomena. This initial number varied between trials because the initial wetting is unsteady and the number of particles adhering during this period is likely dependent on the distance from the channel entrance. Because these particles are not relevant to the study of steady-state particle margination and adhesion propensity, they were subtracted from the total particle adherence data. For each particle size, the number of particles adhered was found to saturate over time. This saturation trend was consistent across trials as well as with that observed in previous particle deposition studies [16,17]. After some time of sustained new particle deposition, the deposited particles become populated on the wall surface, preventing new particles from adhering to the wall because of the near-wall and interparticle repulsive force of these negatively charged particles, as discussed below.

Figure 3(b) shows the number of particles adhered to the top wall normalized by the number of particles in the margination volume (Na/Nmar), which is defined as the volume of fluid in the channel section above (or below) the 55 × 120 μm2 imaged region of the flat wall, as a function of time for a constant particle number concentration of n = (1.0 ± 0.2) × 109 ml−1. Figure 3(c) shows the volume deposition normalized by the volume of particles in the margination volume (Va/Vmar) for a constant particle volume concentration ϕ = 3.1 ± 0.6 vol. % in the flow. Data for 970 nm particles are not included in Figs. 3(b) and 3(c) because no adhesion was observed on the top wall. An additional set of parameters of interest, ΔNsat and ΔVsat, are the steady-state saturation values for the normalized number and volume of particles adhered to the channel wall after 360 s, respectively. The values of ΔNsat and ΔVsat are plotted in Figs. 3(d) and 3(e) for different cases. Each particle size was flown, imaged, and quantified four times except for the 60 and 93 nm particles, which were measured six times each. The error bars represent the random uncertainty exhibited in the multiple measurement results with a 95% confidence interval. For the case of a constant particle number concentration, the percentage of particles adhered to the top wall shows a d−3 power law dependence on the particle diameter d, as shown in Fig. 3(d). However, for the case of constant volume concentration no discernable trend can be observed from the result in Fig. 3(d). In addition, Fig. 3(e) shows no clear diameter dependence in the volume of particles adhered to the wall for either a constant particle number concentration or a constant particle volume concentration at steady state.

Figure 4(a) shows the total number of particles adhered to the flat side of the channel when it constitutes the bottom (B) and top (T) surface of the microchannel, respectively. The data are for all six nanoparticle sizes considered in this study, each after 360 s of flow. The 970 nm particles were found to adhere only to the bottom wall but not to the top wall. In comparison, the difference in adhesion to the top and bottom walls is within the measurement uncertainty for smaller particles. Figure 4(b) suggests that total particle deposition is not affected significantly for the range of shear stresses for any of the particle sizes tested.

Discussion

We first examine the observed difference in total particle deposition between the top and bottom walls for the largest particle size of 970 nm. The relevant forces that induce or retard margination for submicron particles far away from the wall include hydrodynamic, gravitational, Brownian, electrostatic, and van der Waals forces. The hydrodynamic force acts to keep particles on streamlines and is given as [6]
Display Formula

(2)FH=52πμd2γ·

where μ is the dynamic viscosity and γ· is the shear rate. The gravitational force acting perpendicular to the horizontal channel wall is given as [6]
Display Formula

(3)FG=16πd3ρpg

where ρp is the difference in density between the particle and water, and g is the acceleration due to gravity. The Brownian force results in random motion of the particles and is given as [6]
Display Formula

(4)FB=2kBTd

with kB and T being the Boltzmann constant and temperature, respectively. The van der Waals force between two spheres of the same size is approximated as
Display Formula

(5)FVdW=Ad24s2

where s is the separation distance between particles in solution and A is the Hamaker constant for polystyrene spheres separated by water and is equal to 0.95 × 10−20 [32,33]. Coulomb's law gives the electrostatic force between two charges as [34]
Display Formula

(6)Fc=q1q24πɛ0h2

where ε0 is the vacuum permittivity, q is the charge, and h is the separation distance. The charge of a sphere of diameter (d), zeta potential ζ, Debye layer thickness δ, and dielectric constant εr is given by the following expression [34]:
Display Formula

(7)q=ζ[δ+δ/2δ]2πɛ0ɛrd

In Fig. 5(a), these forces are plotted as a function of the diameter of submicron polystyrene spheres for a shear rate of 500 s−1. For particles with diameters of 780 nm and smaller, the Brownian force, responsible for isotropic particle margination, dominates over the gravity force that drives particle sedimentation as well as van der Waals and electrostatic repulsion. Hence, for these small particles, we have found little difference between the particle margination to the top wall and bottom walls. The gravity force increases and the Brownian force decreases with increasing diameter, and become comparable to each other at a diameter of about 1 μm. Because of a large gravity force, the 970 nm particles were found to adhere much more to the bottom wall than the top wall. The d1.25 and d1.3 dependences reported by Gentile et al. [15] and Decuzzi et al. [14] for the percentage of large particles delivered to the bottom wall can also be attributed to this gravity effect.

Electrostatic forces repel particles from each other and result in increased lateral migration of particles. A calculation of the distance between particles in the flow reveals that the average distance would be approximately 10 μm at n = (1.0 ± 0.2) × 109 ml−1. Based on the constant n value and the average zeta potential of −42 mV, the interparticle electrostatic force increases with particle diameter. However, the calculated electrostatic force between adjacent particles in the flow is still much smaller than the Brownian, gravitational, and hydrodynamic forces for the particle diameter range studied here. Moreover, van der Waals interactions are also shown to be several orders of magnitude below all other forces considered here and thus do not affect the margination propensity of the submicron particles in this study.

The d−3 dependence in particle margination for the constant particle number concentration case can be attributed to a dominant Brownian force, which is inversely proportional to d. However, consideration of only the five in-flow forces shown in Fig. 5(a) is insufficient to explain the lack of a clear diameter dependence in the percentage of particles adhered for the case of a constant particle volume fraction (Fig. 3(e)), as well as in the volume of particles adhered for either a constant particle number concentration or a constant particle volume concentration (Fig. 3(f)).

Hence, other forces need to be considered, especially those affecting the near-wall attachment behavior of submicron particles. These forces may include van der Waals forces, colloidal fluid forces, and other electrostatic interactions between particles in the flow and on the wall.

Modeling the interparticle repulsion, we have used the average zeta potential value of −42 mV to calculate and plot in Fig. 5(b), the electrostatic force between a particle in the fluid and adhered particles on the wall as a function of separation distance (h) between the marginating particle and the wall. The calculation is based on the assumption that a new particle marginates into the most energetically favorable location on a uniform bed of adhered particles arranged into a square lattice, as shown in Fig. 6. The force plotted in Fig. 5(b) is the cumulative electrostatic repulsive force between the settling particle and its first, second, and third nearest neighbors among the adhered particles. With the volume of particles adhered on the wall in the imaging area kept constant at 2.28 μm3 of particles per 6600 μm2 of wall area, Fig. 5(b) shows that the repulsive electrostatic force acting on a marginating particle increases with decreasing diameter when the particle is close to the wall, although an opposite diameter dependence is observed when the particle is far away from the wall.

The nearest neighbor distance between the 60 nm particles attached to the wall is as small as 0.5 μm, which is much smaller than the interparticle distance of ∼10 μm in the flow under a constant particle concentration of n = (1.0 ± 0.2) × 109 ml−1. Hence, Figs. 5(a) and 5(b) show that the electrostatic force between a marginating particle and adhered particles is 5 orders and 1 order of magnitude larger than electrostatic repulsive force between adjacent particles in the flow for 60 nm particles and 970 nm particles, respectively, when the marginating particle is within a particle diameter from the wall. Within this distance, the magnitude of the near-wall repulsive force preventing adhesion is comparable to the Brownian force driving the particle margination. Hence, although Brownian force drives the small particles toward the wall, the repulsive electrostatic force from the adhered particles on the wall increases with decreasing particle diameter when the volume of the adhered particles is the same. This competition results in the lack of a clear diameter dependence of the volume of the particles adhered on the wall.

Conclusion

Our experimental results show that the gravity force results in preferential margination of polystyrene spheres of 970 nm diameter to the bottom wall, whereas the margination of smaller polystyrene spheres is isotropic because the Brownian force driving particle margination dominates over the gravity force. The Brownian force increases with decreasing particle diameter, resulting in increasing particle margination with decreasing particle diameter. However, for the same volume of particles adhered on the wall and the same zeta potential on the particles, the repulsive electrostatic force between a near-wall particle in the fluid and the particles adhered on the wall increases with decreasing diameter, and prevents adhesion of small particles. Because of the competition between the Brownian force and the electrostatic force, no clear diameter dependence of the volume of particles adhered to the top wall is observed, which give rises to the observed d−3 dependence in the number of adhered particles. These results reveal the importance of electrostatic force in not only particle endocytosis [35], but also in the margination dynamics near the channel wall. Based on this finding, variation of zeta potential on the particle and wall surface could have been responsible for the discrepancies between different reports including the current work. These results suggest a future direction for a detailed study into the effects of surface charges on both the margination dynamics and internalization behaviors of NCs of different sizes and shapes.

Acknowledgements

This work is supported by National Science Foundation (NSF) Award No. CMMI-0900715. The microfabricaiton was carried out in Microelectronic Research Center, a node in the National Nanofabrication Infrastructure Network supported by NSF. Particle imaging took place at the Institute for Cellular and Molecular Biology (ICMB) at the University of Texas at Austin.

Figures

(a) Cross-sectional scanning electron microscopy (SEM) image of 55 × 35 μm2 channel used in experiments. (b) Fluorescence images taken at the region of interest, designated by the yellow rectangle, on the top channel wall after 93 nm polystyrene spheres were flown in the channel for 20 s (left) and 360 s (right). (C) Time dependent intensity profiles across the top channel wall in the region of interest perpendicular to the flow direction. (D) Total number of particles adhered to the top channel wall in the region of interest at each 20 s intervals based on the measured intensity for a single experiment.

(a) Representative fluorescence images of the top channel wall after 360 s of continuous particle flow in 55-μm-wide channels for each nanoparticle size tested at a shear stress of 7.5 Pa. (b) The number of adhered particles (ΔNsat) normalized by the number of particles in the margination volume as a function of time for a constant particle number concentration of n = (1.0 ± 0.2) × 109 ml−1. (c) The volume of adhered particles (ΔVsat) normalized by the volume of particles in the margination volume as a function of time for a constant particle volume concentration of ϕ = 3.1 ± 0.6 vol. %. (d) Percentage of particles adhered as a function of diameter for constant n = (1.0 ± 0.2) × 109 ml−1 (filled symbols) or constant ϕ = 3.1 ± 0.6 vol. % (unfilled symbols). The data for constant n = (1.0 ± 0.2) × 109 ml−1 can be fitted with a relation of Na/Nmar = 7 × 109 d−3 (line). (e) The volume of particles adhered to the 55 × 120 μm2 imaged section of the top wall as a function of diameter for constant n = (1.0 ± 0.2) × 109 ml−1 and constant ϕ = 3.1 ± 0.6 vol. %. For all plots, the error bars indicate the random uncertainty with a confidence interval of 95%.

(a) Total particle adhesion for the top and bottom walls after 360 s of continuous flow. The bars above “T” are the data for the top wall while those above “B” are the data for the bottom wall. (b) Total particle adhesion of different sized nanoparticles to the top wall of the microchannel at different flow rates maintaining constant particle concentration of 1 × 109 particles/ml.

(a) The hydrodynamic, Brownian, gravitational, van der Waals, and electrostatic forces as a function of diameter for submicron polystyrene spheres in DI water at a shear rate of 500 s−1 at a constant particle number concentration of 1 × 109 particles/ml. The electrostatic force shown is that between adjacent particles in the flow. (b) Electrostatic repulsive force between a marginating particle in the fluid and the adhered particles on the wall when the volume of the adhered particles is taken to be 3.46 × 10−4 m3 adhered per m2 channel wall area and zeta potential is taken to be −42 mV.

Schematic of a particle approaching the channel wall saturated with adhered particles arranged in a square lattice. The nearest neighbor distance of the adhered particles is L. The smallest distance between the marginating particle and the channel wall is h. The distance between the marginating particle and an adhered particle is r.

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